Problem 1068

Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely vertex-connected?

Status

Open

No claim settles this problem.

BoPi24: N. Bowler and M. Pitz, A note on uncountably chromatic graphs. arXiv:2402.05984 (2024).
ErHa66: Erdős, P. and Hajnal, A., On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. (1966), 61-99.
So15: Soukup, Dániel T., Trees, ladders and graphs. J. Combin. Theory Ser. B (2015), 96-116.
Va99: Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).